Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

. We study the semidefinite programming problem (SDP), i.e the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as is specialized to SDP. Next we present an interior point algorithm which converges to the optimal solution in polynomial time. The approach is a direct extension of Yes projective method for linear programming. We also argue that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the significance of these results in a variety of combinatorial optimizationproblems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graphs, and v...